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Abstract

Near-Infrared (NIR) tomographic image reconstruction is a non-linear, ill-posed and ill-conditioned problem, and so in this study, different ways of penalizing the objective function with structural information were investigated. A simple framework to incorporate structural priors is presented, using simple weight matrices that have either Laplacian or Helmholtz-type structures. Using both MRI-derived breast geometry and phantom data, a systematic and quantitative comparison was performed with and without spatial priors. The Helmholtz-type structure can be seen as a more generalized approach for incorporating spatial priors into the reconstruction scheme. Moreover, parameter reduction (i.e. hard prior information) in the imaging field through the enforcement of spatially explicit regions may lead to erroneous results with imperfect spatial priors.

Figures (5)

(a) Simulated μa and μ´s distributions from a breast (obtained from a volunteer) are shown in the first column. Optical properties for the region labeled ‘0’ (fat) are: μa = 0.006 mm-1 and μs = 0.6 mm-1. Region ‘1’ (fibroglandular) values are: μa = 0.012 mm-1 and μ´s = 1.2 mm-1. Region ‘2’ (tumor) values are: μa = 0.018 mm-1 and μ´s = 1.8 mm-1. Reconstructed μa and μ´s images from different techniques with simulated data having 1% random noise and imperfect structural information in defining region ‘1’ (7% reduction compared to the original segmentation) are shown in the rest of the columns. The middle two columns use soft prior structural information while the last column shows the result with hard prior information. In the Helmholtz case, κ = 1/8 mm-1 (BPE) was used. (b) Cross-sectional plots, along the dotted line in the actual image (see first column of (a)), of true and reconstructed μa and μ´s distributions.

(a) Reconstructed μa and μ´s images from different techniques with simulated data having 5% random noise and perfect structural priors (actual images are shown in the first column of Fig. 1(a)). The first column shows the reconstruction results without the use of prior information. The middle two columns use soft prior structural information while the last row shows the result with hard prior information. In the Helmholtz case, κ = 1/8 mm-1 (BPE) was used. (b) The mean values and standard deviations (plotted as error bars) in μa and μ´s for different regions of breast geometry (labeled in actual image) with increasing noise level (1% to 10 %).

(a) Reconstructed μa and μ´s images from the experimental phantom data using Helmholtz-type regularization matrix for different values of κ, which are given at the top of each column. (b) Cross-sectional plots along the dotted line of the actual images in Fig. 4(a) (first column) are shown with the data from reconstructed μa and μ´s images in (a). The best prior estimate (BPE) case (κ = 1/16 mm-1) is also presented for comparison.

Tables (2)

Table 1. Mean and standard deviation of the reconstructed (a) μa and (b) μ´s values in different regions (labeled in first column of Fig. 1(a)) recovered with simulated data having 1% random noise and imperfect structural information defining region ‘1’ (7% reduction compared to the original segmentation). The corresponding reconstructed images are shown in Fig. 1(a)

Table 2. Mean and standard deviation of the reconstructed (a) μa and (b) μ´s values in different regions (labeled in first column of Fig. 4(a)) recovered from the experimental phantom data. The corresponding reconstructed images are shown in Fig. 4(a) and 5(a).

Metrics

Table 1.

Mean and standard deviation of the reconstructed (a) μa and (b) μ´s values in different regions (labeled in first column of Fig. 1(a)) recovered with simulated data having 1% random noise and imperfect structural information defining region ‘1’ (7% reduction compared to the original segmentation). The corresponding reconstructed images are shown in Fig. 1(a)

Methods

Region-0

Region-1

Region-2

Actual

0.006

0.012

0.018

Laplacian

0.0064±0.0010

0.0117±0.0018

0.0156±0.0018

Helmholtz (κ = 1/8)

0.0062±0.0011

0.0120±0.0020

0.0156±0.0017

Hard Priors

0.006

0.0118

0.0843

Table 2.

Mean and standard deviation of the reconstructed (a) μa and (b) μ´s values in different regions (labeled in first column of Fig. 4(a)) recovered from the experimental phantom data. The corresponding reconstructed images are shown in Fig. 4(a) and 5(a).

Methods

Region-0

Region-1

Region-2

Actual

0.0065

0.01

0.02

No Priors

0.0025±0.0010

0.0045±0.0022

0.0120±0.0090

Laplacian

0.0031±0.0002

0.0051±0.0005

0.0174±0.0029

Helmholtz (κ = 1/16)

0.0015±0.0005

0.0058±0.0009

0.0241±0.0043

Hard Priors

0.0032

0.005

0.0213

Helmholtz (κ = 1/5)

0.0009±0.0006

0.0061±0.0008

0.0191±0.0031

Helmholtz (κ = 1/43)

0.0027±0.0003

0.0052±0.0007

0.0234±0.0043

Helmholtz (κ = 1/86)

0.0022±0.0005

0.0061±0.0032

0.0192±0.0044

Tables (2)

Table 1.

Mean and standard deviation of the reconstructed (a) μa and (b) μ´s values in different regions (labeled in first column of Fig. 1(a)) recovered with simulated data having 1% random noise and imperfect structural information defining region ‘1’ (7% reduction compared to the original segmentation). The corresponding reconstructed images are shown in Fig. 1(a)

Methods

Region-0

Region-1

Region-2

Actual

0.006

0.012

0.018

Laplacian

0.0064±0.0010

0.0117±0.0018

0.0156±0.0018

Helmholtz (κ = 1/8)

0.0062±0.0011

0.0120±0.0020

0.0156±0.0017

Hard Priors

0.006

0.0118

0.0843

Table 2.

Mean and standard deviation of the reconstructed (a) μa and (b) μ´s values in different regions (labeled in first column of Fig. 4(a)) recovered from the experimental phantom data. The corresponding reconstructed images are shown in Fig. 4(a) and 5(a).